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Properties

Label 24-112e12-1.1-c9e12-0-0
Degree $24$
Conductor $3.896\times 10^{24}$
Sign $1$
Analytic cond. $1.35726\times 10^{21}$
Root an. cond. $7.59499$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.78e4·9-s + 1.04e7·25-s − 3.24e6·29-s + 1.66e7·37-s + 5.36e7·49-s − 2.65e8·53-s + 2.25e9·81-s + 3.47e9·109-s + 2.58e9·113-s + 1.09e10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6.40e10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 3.95·9-s + 5.33·25-s − 0.852·29-s + 1.45·37-s + 1.32·49-s − 4.61·53-s + 5.82·81-s + 2.35·109-s + 1.48·113-s + 4.62·121-s + 6.04·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{48} \cdot 7^{12}\)
Sign: $1$
Analytic conductor: \(1.35726\times 10^{21}\)
Root analytic conductor: \(7.59499\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{48} \cdot 7^{12} ,\ ( \ : [9/2]^{12} ),\ 1 )\)

Particular Values

\(L(5)\) \(\approx\) \(0.5958308578\)
\(L(\frac12)\) \(\approx\) \(0.5958308578\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - 7658922 p T^{2} + 1923939993 p^{7} T^{4} - 5211714787220 p^{12} T^{6} + 1923939993 p^{25} T^{8} - 7658922 p^{37} T^{10} + p^{54} T^{12} \)
good3 \( ( 1 + 38942 T^{2} + 127296047 p^{2} T^{4} + 365850312452 p^{4} T^{6} + 127296047 p^{20} T^{8} + 38942 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
5 \( ( 1 - 5212578 T^{2} + 15308012208471 T^{4} - 258750382129022252 p^{3} T^{6} + 15308012208471 p^{18} T^{8} - 5212578 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
11 \( ( 1 - 5456494686 T^{2} + 12704777856531536343 T^{4} - \)\(21\!\cdots\!96\)\( T^{6} + 12704777856531536343 p^{18} T^{8} - 5456494686 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
13 \( ( 1 - 32048162706 T^{2} + \)\(66\!\cdots\!35\)\( T^{4} - \)\(82\!\cdots\!56\)\( T^{6} + \)\(66\!\cdots\!35\)\( p^{18} T^{8} - 32048162706 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
17 \( ( 1 - 203178352230 T^{2} + \)\(20\!\cdots\!43\)\( T^{4} - \)\(59\!\cdots\!92\)\( p^{2} T^{6} + \)\(20\!\cdots\!43\)\( p^{18} T^{8} - 203178352230 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
19 \( ( 1 + 794033961918 T^{2} + \)\(39\!\cdots\!11\)\( T^{4} + \)\(13\!\cdots\!40\)\( T^{6} + \)\(39\!\cdots\!11\)\( p^{18} T^{8} + 794033961918 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
23 \( ( 1 - 10034999249190 T^{2} + \)\(43\!\cdots\!27\)\( T^{4} - \)\(10\!\cdots\!36\)\( T^{6} + \)\(43\!\cdots\!27\)\( p^{18} T^{8} - 10034999249190 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
29 \( ( 1 + 811902 T + 23395248825315 T^{2} + 37038808640745674868 T^{3} + 23395248825315 p^{9} T^{4} + 811902 p^{18} T^{5} + p^{27} T^{6} )^{4} \)
31 \( ( 1 + 76484368738170 T^{2} + \)\(38\!\cdots\!23\)\( T^{4} + \)\(11\!\cdots\!48\)\( T^{6} + \)\(38\!\cdots\!23\)\( p^{18} T^{8} + 76484368738170 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
37 \( ( 1 - 4150314 T + 314725876292187 T^{2} - \)\(12\!\cdots\!76\)\( T^{3} + 314725876292187 p^{9} T^{4} - 4150314 p^{18} T^{5} + p^{27} T^{6} )^{4} \)
41 \( ( 1 - 307196916375174 T^{2} + \)\(12\!\cdots\!23\)\( T^{4} - \)\(84\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!23\)\( p^{18} T^{8} - 307196916375174 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
43 \( ( 1 + 463029279593154 T^{2} + \)\(51\!\cdots\!19\)\( T^{4} + \)\(12\!\cdots\!20\)\( T^{6} + \)\(51\!\cdots\!19\)\( p^{18} T^{8} + 463029279593154 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
47 \( ( 1 + 5845916462833434 T^{2} + \)\(15\!\cdots\!83\)\( T^{4} + \)\(21\!\cdots\!80\)\( T^{6} + \)\(15\!\cdots\!83\)\( p^{18} T^{8} + 5845916462833434 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
53 \( ( 1 + 66255342 T + 3798061250060715 T^{2} + \)\(28\!\cdots\!72\)\( T^{3} + 3798061250060715 p^{9} T^{4} + 66255342 p^{18} T^{5} + p^{27} T^{6} )^{4} \)
59 \( ( 1 + 30567639856197774 T^{2} + \)\(47\!\cdots\!83\)\( T^{4} + \)\(48\!\cdots\!04\)\( T^{6} + \)\(47\!\cdots\!83\)\( p^{18} T^{8} + 30567639856197774 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
61 \( ( 1 - 37385357617080690 T^{2} + \)\(78\!\cdots\!95\)\( T^{4} - \)\(11\!\cdots\!12\)\( T^{6} + \)\(78\!\cdots\!95\)\( p^{18} T^{8} - 37385357617080690 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
67 \( ( 1 - 92666385687376878 T^{2} + \)\(50\!\cdots\!95\)\( T^{4} - \)\(16\!\cdots\!72\)\( T^{6} + \)\(50\!\cdots\!95\)\( p^{18} T^{8} - 92666385687376878 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
71 \( ( 1 - 47397823203564966 T^{2} + \)\(33\!\cdots\!83\)\( T^{4} - \)\(32\!\cdots\!96\)\( p T^{6} + \)\(33\!\cdots\!83\)\( p^{18} T^{8} - 47397823203564966 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
73 \( ( 1 - 231540780219491526 T^{2} + \)\(28\!\cdots\!47\)\( T^{4} - \)\(20\!\cdots\!88\)\( T^{6} + \)\(28\!\cdots\!47\)\( p^{18} T^{8} - 231540780219491526 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
79 \( ( 1 - 311745936164247318 T^{2} + \)\(67\!\cdots\!51\)\( T^{4} - \)\(95\!\cdots\!20\)\( T^{6} + \)\(67\!\cdots\!51\)\( p^{18} T^{8} - 311745936164247318 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
83 \( ( 1 + 400169799553742814 T^{2} + \)\(14\!\cdots\!99\)\( T^{4} + \)\(28\!\cdots\!60\)\( T^{6} + \)\(14\!\cdots\!99\)\( p^{18} T^{8} + 400169799553742814 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
89 \( ( 1 - 1395315081290199846 T^{2} + \)\(10\!\cdots\!03\)\( T^{4} - \)\(44\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!03\)\( p^{18} T^{8} - 1395315081290199846 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
97 \( ( 1 - 2330410526637666438 T^{2} + \)\(33\!\cdots\!27\)\( T^{4} - \)\(29\!\cdots\!40\)\( T^{6} + \)\(33\!\cdots\!27\)\( p^{18} T^{8} - 2330410526637666438 p^{36} T^{10} + p^{54} T^{12} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.02875242271073377491997907192, −3.02605800889126059462067304688, −2.90007124214799828101943151809, −2.83913970524578915645369762860, −2.68538328956372483575653417878, −2.65355493915312194984036665277, −2.21897033975595802903637499371, −2.19294598439212079991398745407, −2.09736391520940589642330313050, −2.02013903276369998330440181561, −2.01721698027747264150075208104, −1.74140820078005375174256228920, −1.68554084853983016861225891398, −1.46870395604250175064650086190, −1.24279511418198955158742994636, −1.04662241619662367544451024565, −0.964142382906974562255019138237, −0.955259392360629719992742344500, −0.870364078575986041019675190817, −0.792862166572124194039907469275, −0.54768943175238389360795356360, −0.48394301570325391221366461654, −0.25551781082426368725062952573, −0.12370040426798215454213897694, −0.07264240409865662307067801800, 0.07264240409865662307067801800, 0.12370040426798215454213897694, 0.25551781082426368725062952573, 0.48394301570325391221366461654, 0.54768943175238389360795356360, 0.792862166572124194039907469275, 0.870364078575986041019675190817, 0.955259392360629719992742344500, 0.964142382906974562255019138237, 1.04662241619662367544451024565, 1.24279511418198955158742994636, 1.46870395604250175064650086190, 1.68554084853983016861225891398, 1.74140820078005375174256228920, 2.01721698027747264150075208104, 2.02013903276369998330440181561, 2.09736391520940589642330313050, 2.19294598439212079991398745407, 2.21897033975595802903637499371, 2.65355493915312194984036665277, 2.68538328956372483575653417878, 2.83913970524578915645369762860, 2.90007124214799828101943151809, 3.02605800889126059462067304688, 3.02875242271073377491997907192

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.